Optimal. Leaf size=90 \[ \frac {4 c \sqrt {b x+c x^2} (5 b B-4 A c)}{15 b^3 x}-\frac {2 \sqrt {b x+c x^2} (5 b B-4 A c)}{15 b^2 x^2}-\frac {2 A \sqrt {b x+c x^2}}{5 b x^3} \]
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Rubi [A] time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 658, 650} \begin {gather*} \frac {4 c \sqrt {b x+c x^2} (5 b B-4 A c)}{15 b^3 x}-\frac {2 \sqrt {b x+c x^2} (5 b B-4 A c)}{15 b^2 x^2}-\frac {2 A \sqrt {b x+c x^2}}{5 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \sqrt {b x+c x^2}} \, dx &=-\frac {2 A \sqrt {b x+c x^2}}{5 b x^3}+\frac {\left (2 \left (-3 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{5 b}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{5 b x^3}-\frac {2 (5 b B-4 A c) \sqrt {b x+c x^2}}{15 b^2 x^2}-\frac {(2 c (5 b B-4 A c)) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{15 b^2}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{5 b x^3}-\frac {2 (5 b B-4 A c) \sqrt {b x+c x^2}}{15 b^2 x^2}+\frac {4 c (5 b B-4 A c) \sqrt {b x+c x^2}}{15 b^3 x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 0.60 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} \left (A \left (3 b^2-4 b c x+8 c^2 x^2\right )+5 b B x (b-2 c x)\right )}{15 b^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 60, normalized size = 0.67 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (3 A b^2-4 A b c x+8 A c^2 x^2+5 b^2 B x-10 b B c x^2\right )}{15 b^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 57, normalized size = 0.63 \begin {gather*} -\frac {2 \, {\left (3 \, A b^{2} - 2 \, {\left (5 \, B b c - 4 \, A c^{2}\right )} x^{2} + {\left (5 \, B b^{2} - 4 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, b^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 133, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B \sqrt {c} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b \sqrt {c} + 3 \, A b^{2}\right )}}{15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (8 A \,c^{2} x^{2}-10 B b c \,x^{2}-4 A b c x +5 B \,b^{2} x +3 A \,b^{2}\right )}{15 \sqrt {c \,x^{2}+b x}\, b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 106, normalized size = 1.18 \begin {gather*} \frac {4 \, \sqrt {c x^{2} + b x} B c}{3 \, b^{2} x} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{2}}{15 \, b^{3} x} - \frac {2 \, \sqrt {c x^{2} + b x} B}{3 \, b x^{2}} + \frac {8 \, \sqrt {c x^{2} + b x} A c}{15 \, b^{2} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{5 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 56, normalized size = 0.62 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (5\,B\,b^2\,x+3\,A\,b^2-10\,B\,b\,c\,x^2-4\,A\,b\,c\,x+8\,A\,c^2\,x^2\right )}{15\,b^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{3} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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